Alexander polynomial, finite type invariants and volume of hyperbolic knots

Kalfagianni, Efstratia¹

¹Department of Mathematics, Michigan State University, East Lansing MI 48824, USA, School of Mathematics, Institute for Advanced Study, Princeton NJ 08540, USA

Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 1111-1123

Résumé

We show that given n > 0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order ≤ n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the complement of a hyperbolic alternating knot is bounded above by a linear function of the coefficients of the Alexander polynomial of the knot. As a corollary to our main result we obtain that, for every m > 0, there exists a sequence of hyperbolic knots with trivial finite type invariants of order ≤ m but arbitrarily large volume. We discuss how our results fit within the framework of relations between the finite type invariants and the volume of hyperbolic knots, predicted by Kashaev’s hyperbolic volume conjecture.

Détail

DOI : 10.2140/agt.2004.4.1111

Keywords: Alexander polynomial, finite type invariants, hyperbolic knot, hyperbolic Dehn filling, volume.

Bibliographie

[1] C C Adams, Volumes of $N$–cusped hyperbolic 3–manifolds, J. London Math. Soc. $(2)$ 38 (1988) 555

[2] N Askitas, E Kalfagianni, On knot adjacency, Topology Appl. 126 (2002) 63

[3] D Bar-Natan, S Garoufalidis, On the Melvin–Morton–Rozansky conjecture, Invent. Math. 125 (1996) 103

[4] F Bonahon, Geometric structures on 3–manifolds, from: "Handbook of geometric topology", North-Holland (2002) 93

[5] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter Co. (1985)

[6] P J Callahan, A W Reid, Hyperbolic structures on knot complements, Chaos Solitons Fractals 9 (1998) 705

[7] O T Dasbach, X S Lin, A volumish theorem for the Jones polynomial of alternating knots, Pacific J. Math. 231 (2007) 279

[8] D Gabai, Foliations and the topology of 3–manifolds III, J. Differential Geom. 26 (1987) 479

[9] H Howards, J Luecke, Strongly $n$–trivial knots, Bull. London Math. Soc. 34 (2002) 431

[10] E Kalfagianni, X S Lin, Knot adjacency and satellites, Topology Appl. 138 (2004) 207

[11] E Kalfagianni, X S Lin, Knot adjacency, genus and essential tori, Pacific J. Math. 228 (2006) 251

[12] R M Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269

[13] W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37

[14] H Murakami, J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85

[15] D S Silver, S G Williams, Mahler measure of Alexander polynomials, J. London Math. Soc. $(2)$ 69 (2004) 767

[16] D S Silver, A Stoimenow, S G Williams, Euclidean Mahler measure and twisted links, Algebr. Geom. Topol. 6 (2006) 581

[17] A Stoimenow, Graphs, determinants of knots and hyperbolic volume, Pacific J. Math. 232 (2007) 423

[18] W Thurston, The geometry and topology of 3–manifolds